Question

Vă rog mult!!!!!!!!!

Answer 1

4)
[tex]C_n^1+C_n^2=15\\ \\ \frac{n!}{1!(n-1)!}+\frac{n!}{2!(n-2)!}=15\\ \\ \frac{(n-1)!\cdot n}{(n-1)!}+\frac{(n-2)!\cdot(n-1)\cdot n}{2(n-2)!}=15\\ \\ n+\frac{1}{2}(n-1)n=15\\ \\ 2n+n(n-1)=30\\ n^2-n+2n-30=0\\ n^2+n-30=0\\ \Delta=1-4\cdot-3=121\Rightarrow\sqrt{\Delta}=11\\ n_{1,2}=\frac{-1\pm11}{2}\\ n_1=\frac{-1+11}{2}=\frac{10}{2}=5\\ n_2=\frac{-1-11}{2}\ \textless \ 0 [/tex]
Deci, $n=5\in\mathbb{N}, n \geq 2$
5)
Un punct se afla pe o dreapta daca, coordonatele punctului verifica ecuatia dreptei.
[tex]d:x-y+1=0 \\A_m(2m-1,m^2)\in d\Leftrightarrow 2m-1-m^2+1=0\Leftrightarrow -m^2+2m=0\Leftrightarrow \\ m^2-2m=0\Leftrightarrow m(m-2)=0\Leftrightarrow(m=0\: sau\: m=2)[/tex]
Deci, 
$m\in\{0,\:2\}$